Intertemporal Demand Allocation for Inventory Control in Online Marketplaces

Rene Caldentey; Tong Xie

cs.MAApr 8, 2026

Intertemporal Demand Allocation for Inventory Control in Online Marketplaces

Authors: Rene Caldentey, Tong Xie

Organizations: Booth School of Business, The University of Chicago, Chicago, IL 60637

Abstract

Online marketplaces increasingly do more than simply match buyers and sellers: they route orders across competing sellers and, in many categories, offer ancillary fulfillment services that make seller inventory a source of platform revenue. We investigate how a platform can use intertemporal demand allocation to influence sellers' inventory choices without directly controlling stock. We develop a model in which the platform observes aggregate demand, allocates orders across sellers over time, and sellers choose between two fulfillment options, fulfill-by-merchant (FBM) and fulfill-by-platform (FBP), while replenishing inventory under state-dependent base-stock policies. The key mechanism we study is informational: by changing the predictability of each seller's sales stream, the platform changes sellers' safety-stock needs even when average demand shares remain unchanged. We focus on nondiscriminatory allocation policies that give sellers the same demand share and forecast risk. Within this class, uniform splitting minimizes forecast uncertainty, whereas any higher level of uncertainty can be implemented using simple low-memory allocation rules. Moreover, increasing uncertainty above the uniform benchmark requires routing rules that prevent sellers from inferring aggregate demand from their own sales histories. These results reduce the platform's problem to choosing a level of forecast uncertainty that trades off adoption of platform fulfillment against the inventory held by adopters. Our analysis identifies demand allocation as a powerful operational and informational design lever in digital marketplaces.

Figures & tables

Figure 1: Overview of the platform marketplace. Aggregate demand $D_{t}$ is allocated across sellers, who choose between FBM and FBP and manage their inventory stocks using state-dependent base-stock policies.

Panel of Figure 2 — Figure 2: Partition of sellers (dots) in the $(\Delta K,\Delta F)$ plane under a neutral policy $\pi$ . Sellers above $\Delta F=\frac{N\sigma^{\pi}}{\mu}\Delta K$ adopt FBP (shaded region), while sellers below prefer FBM.

Panel of Figure 3 — Figure 3: Left: number of FBP adopters $|N^{\scalebox{0.45}{\mbox{\rm FBP}}}(\pi)|$ . Right: cumulative safety stock of FBP adopters, $\Gamma^{\scalebox{0.45}{\mbox{\rm FBP}}}(\pi)$ .

Panel of Figure 4 — Figure 4: Demand sample paths: Left: equal MSFE, different variances; Right: equal variances, different MSFE.

n_{k}\in\arg\min_{n\in[N]}\bigl\{A_{n}-b_{n,t}\bigr\}

Algorithm 1 Offset-Tracking

Panel of Figure 5 — Figure 5: Geographic trade-off between holding and fulfillment costs (left) and seller cost parameters with safety-stock coefficients (right). Each seller $n$ is located at holding cost $h_{n}$ and fulfillment cost $f_{n}$ ; sellers nearer the city face higher $h_{n}$ but lower $f_{n}$ . The shaded region highlights the urban area containing sellers 8, 9, and 10, who switch from FBP to FBM in the optimal policy.

Panel of Figure 6 — Figure 6: Sellers in the $(\Delta K,\Delta F)$ plane. The shaded cone corresponds to feasible values $\sigma\in[\sigma_{\mbox{\tiny\rm L}},\sigma_{\mbox{\tiny\rm U}}]$ , defined by the boundaries $\Delta F=(N\sigma_{\mbox{\tiny\rm L}}/\mu)\Delta K$ and $\Delta F=(N\sigma_{\mbox{\tiny\rm U}}/\mu)\Delta K$ .

Panel of Figure 7 — Figure 7: Number of FBP adopters (left) and platform payoff $V(\sigma)$ (right) as functions of $\sigma$ . The shaded area corresponds to the feasible region $\sigma\in[\sigma_{\mbox{\tiny\rm L}},\sigma_{\mbox{\tiny\rm U}}]$ , with $\sigma_{\mbox{\tiny\rm L}}=0.5$ and $\sigma_{\mbox{\tiny\rm U}}=33$ .

Demand Allocation	Uniform	Optimal
Root MSFE: $\sigma$	$\sigma_{\mbox{\tiny\rm L}}=0.50$	$\sigma^{\star}=8.87$
FBP adopters: $N^{\scalebox{0.45}{\mbox{\rm FBP}}}(\sigma)$	$\{1,2,3,4,5,6,7,8,9,10\}$	$\{1,2,3,4,5,6,7\}$
Number of FBP adopters: $\big\|N^{\scalebox{0.45}{\mbox{\rm FBP}}}(\sigma)\big\|$	$10$	$7$
Platform payoff: $V(\sigma)$	$293.78$	$372.45$
Cumulative FBP safety stock: $\Gamma^{\scalebox{0.45}{\mbox{\rm FBP}}}(\sigma)$	$4.39$	$52.73$
Cumulative FBM safety stock: $\Gamma^{\scalebox{0.45}{\mbox{\rm FBM}}}(\sigma)$	$0$	$28.12$

Table 1: Outcomes under the uniform allocation

\sigma=\sigma_{\mbox{\tiny\rm L}}

and the platform-optimal choice

\sigma=\sigma^{\star}

E-Commerce

E-commerce studies digital marketplaces, including product discovery, recommendations, pricing, transactions, and AI-assisted shopping and operations.

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Understanding Strategic Platform Entry and Seller Exploration: A Stackelberg Model

Online market platforms play an increasingly powerful role in the economy. An empirical phenomenon is that platforms, such as Amazon, Apple, and DoorDash, also enter their own marketplaces, imitating successful products developed by third-party sellers. We formulate a Stackelberg model, where the platform acts as the leader by committing to an entry policy: when will it enter and compete on a product? We study this model through a theoretical and computational framework. We begin with a single seller, and consider different kinds of policies for entry. We characterize the seller's optimal explore-exploit strategy via a Gittins-index policy, and give an algorithm to compute the platform's optimal entry policy. We then consider multiple sellers, to account for competition and information spillover. Here, the Gittins-index characterization fails, and we employ deep reinforcement learning to examine seller equilibrium behavior. Our findings highlight the incentives that drive platform entry and seller innovation, consistent with empirical evidence from markets such as Amazon and Google Play, with implications for regulatory efforts to preserve innovation and market diversity.

Garrett Seo, Xintong Wang, David C. Parkes

Rutgers University, New Brunswick, NJ, USA · Harvard University, Cambridge, MA, USA

E-Commerce Stackelberg Games Exploration-Exploitation Tradeoff

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May 8, 2026

Online Allocation with Unknown Shared Supply

Many real-world resource allocation systems, such as humanitarian logistics and vaccine distribution, must preposition limited supply across multiple locations before demand is realized while stockouts incur irreversible service losses. To study this, we introduce the Online Shared Supply Allocation (OSSA) problem, a stateful online model in which a central hub allocates a finite, unknown supply to multiple sites facing sequential demand under fixed-charge transportation costs and lost-sales penalties. Unlike classical make-to-stock or make-to-order inventory models, OSSA precludes backlogging and replenishment only hedges against future demand. To tackle OSSA, we propose a deterministic threshold-proportional policy GPA and prove that it achieves a

4/3

-approximation to the offline optimum up to an additive term independent of the total supply. We complement this with matching lower bounds showing that the

4/3

ratio is tight and that the additive-error dependence is unavoidable, even for randomized algorithms that know the total supply upfront. Finally, we develop a learning-augmented extension to GPA that principally incorporates imperfect forecasts (e.g., from human experts or ML models) commonly available in practice, enabling us to exploit high-quality advice while being robust against arbitrary bad ones. Synthetic and real-world experiments show that GPA outperforms natural baselines with global supply is scarce.

Tzeh Yuan Neoh, Davin Choo, Mengchu Yue +1

Harvard University

Supply Chain Management Sequential Decision Making Online Resource Allocation Inventory Control Learning-Augmented Algorithms Approximation Algorithms+1

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May 25, 2026

Optimal and Order-optimal Gated Priority-based Greedy Policies for Two-layer Multi-item Order Fulfillment

We study how an e-commerce firm should make real-time fulfillment decisions in a two-layer distribution network when multi-item customer orders arrive sequentially and future demand is unknown. The central managerial tension is whether to use scarce front distribution center (FDC) inventory to save current fulfillment cost or preserve that inventory for future orders that may be more valuable to serve locally. We formulate an adversarial online model with multiple FDCs, one regional distribution center (RDC), multi-unit multi-item orders, and item-specific and time-varying variable costs. Our theoretical objective is to characterize when simple, interpretable, and implementable fulfillment rules can perform nearly as well as an optimal clairvoyant planner. We develop a family of Gated Priority-based Greedy policies, derive competitive-ratio guarantees under both time-varying and time-invariant cost structures, and establish matching or near-matching lower bounds for any online algorithm. Numerical experiments show that the proposed policies perform strongly relative to generalized myopic and forecast-based benchmarks. The analysis yields managerial guidance on when local inventory should be protected, when splitting orders is worth the fixed-cost burden, and how the relative magnitudes of fixed and variable costs determine the value of more sophisticated optimization.

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Leonard N. Stern School of Business, New York University, New York, NY 10012, USA · Qiuzhen College, Tsinghua University, Beijing 100084, China · Yau Mathematical Sciences Center & Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Inventory Control Competitive Analysis Supply Chain Management E-Commerce Online Resource Allocation Supply Chain Optimization

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